Second order linear ODE: Difference between revisions
(Created page with "Second order linear ODEs are in the following form: <math> y''+p(t)y'+q(t)y=g(t) </math> Important types of second order linear ODEs include * Homogeneous * Constant coefficients (where p and q are constants) = Initial value problem = There are two arbitrary constants in the solution of a second order linear ODE, so we need two initial conditions. <math> \begin{cases} y(t_0)=y_0\\ y'(t_0)=y_0' \end{cases} </math> = Solutions = == Constant coefficient, homogeneou...") |
|||
| Line 31: | Line 31: | ||
<math> | <math> | ||
y=e^{rt} | y=e^{rt} | ||
</math> | |||
We substitute in the guess and obtain the characteristic equation | |||
<math> | |||
\begin{aligned} | |||
ar^2e^{rt}+bre^{rt}+ce^{rt}&=0\\ | |||
ar^2+br+c&=0 | |||
\end{aligned} | |||
</math> | </math> | ||
[[Category:Differential Equations]] | [[Category:Differential Equations]] | ||
Revision as of 21:31, 1 May 2024
Second order linear ODEs are in the following form:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y''+p(t)y'+q(t)y=g(t) }
Important types of second order linear ODEs include
- Homogeneous
- Constant coefficients (where p and q are constants)
Initial value problem
There are two arbitrary constants in the solution of a second order linear ODE, so we need two initial conditions.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} y(t_0)=y_0\\ y'(t_0)=y_0' \end{cases} }
Solutions
Constant coefficient, homogeneous
These are the simplest kind. They have the general form
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ay''+by'+cy=0 }
An exponential function has the property of being the same after many differentiation. We take advantage of this property and guess the solution to be the form of
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=e^{rt} }
We substitute in the guess and obtain the characteristic equation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{aligned} ar^2e^{rt}+bre^{rt}+ce^{rt}&=0\\ ar^2+br+c&=0 \end{aligned} }
